Fracloor alternating harmonic

If $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{5n^2}\int_0^{5n}\int_0^{5n}\left\{y\right\}^{\lfloor x\rfloor} dxdy=\frac{a\pi^2}{b}-\frac{1}{c}\log^2c -c\log^2\phi$ , then find $a+b+2c$ where $a,b,c$ are positive integers with being $a, c$ co-prime numbers.

Notation: $\left\{.\right\}$ is ceiling function $\left\lfloor. \right\rfloor$ is floor function , $\zeta(.)$ is Riemann zeta function and $\phi$ is Golden ratio .

This is an original problem which is extended version of Integration of Weired function-4 and somehow related and advance version of Harmonic sum .